Band-Limited Equivalence of Convolution Operators and its Application to Filtered Vorticity Dynamics

TL;DR

This paper proves that convolution operators with identical Fourier transforms over a finite spectral band are equivalent on band-limited functions and applies this to understand filtered vorticity dynamics, supported by numerical examples.

Contribution

It establishes a general theorem on the equivalence of band-limited convolution operators and applies it to clarify spectral measurement limitations in vorticity dynamics.

Findings

Convolution operators with identical Fourier transforms over a band are equivalent on band-limited functions.

Real-space diagnostics can underestimate spectral proportionality due to unobservable degrees of freedom.

Numerical illustrations support the theoretical results.

Abstract

In this study, we established a general theorem regarding the equivalence of convolution operators restricted to a finite spectral band. We demonstrated that two kernels with identical Fourier transforms over the resolved band act identically on all band-limited functions, even if their kernels differ outside the band. This property is significant in applied mathematics and computational physics, particularly in scenarios where measurements or simulations are spectrally truncated. As an application, we examine the proportionality relation $S(\boldsymbol {r}) \approx \zeta\,\omega(\boldsymbol{r})$ in filtered vorticity dynamics and clarify why real-space diagnostics can underestimate the spectral proportionality due to unobservable degrees of freedom. Our theoretical findings were supported by numerical illustrations using synthetic data.